OR 647: Queueing Theory

School of Information Technology and Engineering

Department of Systems Engineering and Operations Research



OR 647:                  Queueing Theory


                                Dr. Frederick Wieland

Phone: 703-883-5385 (Office – MITRE/CAASD); Email: fwieland@mitre.org

Office Hours: By arrangement with instructor


TEXT:                    Fundamentals of Queueing Theory (3rd Edition)

                                Donald Gross & Carl M. Harris, John Wiley & Sons



QTS, Queueing Theory Software, for use in conjunction with the textbook. The software is available in the format of self-extracting Windows zip files for EXCEL and Quattro Pro 8 for Windows 95, 98 and 2000. The latest is an Enhanced Version of the software called QtsPlus by James M. Thompson, Carl Harris and Donald Gross for Excel 97 and above.


Download the software from ftp://ftp.wiley.com/public/sci_tech_med/queueing_theory/


The URL for the book errata is:





                Mid-term Exam:     40%   (half take-home using software, half in class – open book)

                Final Exam:            40%   (half take-home using software, half in class – open book -- cumulative)

                Homework #1:       10%

                Homework #2:       10%



The intent of this course is to provide a modern perspective on the analysis of stochastic waiting-line systems.  There will be an emphasis on the underlying random processes, ultimately leading to the development of practical strategies for dealing with the design and control of queues in a contemporary technological environment. Prerequisites are a knowledge of the fundamental elements of probability (no background in statistical inference is needed) and a general graduate level maturity in applied math. There will be a special emphasis on the numerical solution of queueing problems using the computational package.



Week                     Topic                                                                                                      Assignments


1/22/02                   Course overview                                                                                  Read Ch 1, G&H  

                                Notation/ Measures of Effectiveness/Lindley’s Equation            Download QTS software

                                Poisson Process and the Exponential Distribution


1/29/02                   Markovian Property of the Exponential Distribution                     Read Ch 1, G&H  

                                Stochastic Processes and Markov Chains

                                Long-run Behavior of Markov Processes


2/5/02                     Birth-Death Processes                                                                        Read Ch 1, G&H

                                Steady-state solution for the M/M/1 model

                                Methods of  solving steady-state difference equations                              


2/12/02                   M/M/1 Measures of Effectiveness and Waiting-time distributions            Read Ch 2, G&H

                                Queues with Parallel Channels (M/M/c)                                                         


2/19/02                   Queues with Parallel Channels (M/M/c)  (cont)                             Read Ch 2, G&H

                                Queues with Parallel Channels and Truncation (M/M/c/K)         Assign. #1 given

                                Erlang’s Formula (M/G/c/c)


2/26/02                   Queues with Unlimited Service                                                          Read Ch 2, G&H

Finite-Source Queues

State-Dependent Service                                   

Queues with Impatience


3/5/02                     Transient Solution to M/M/1                                                             Read Ch 3, G&H

Bulk Input/ Bulk Service                                                                     Assign. #1 due

                                                                                                                Takehome midterm given   


3/12/02                   Erlangian Models


3/19/02                   Spring Break                                                                                        


3/26/02                   MIDTERM: In-class portion                                                              Takehome midterm due


4/2/02                     Series Queues                                                                                       Read Ch 4, G&H

                                Open Jackson Networks/ Closed Jackson Networks


4/9/02                     Jackson Networks                                                                                Read Ch 5, G&H

                                M/G/1 queues                                                                                       Assign. #2 given


4/16/02                   G/M/1 queues                                                                                       Read Ch 6.2,6.3,6.6 G&H

                                G/G/1 queues


4/23/02                   Matrix Geometric Methods and Call Center Example                     Assign. #2 due

                                M/D/c queues                                                                                      Read 7.1,7.2,7.4 G&H          

Design of queues


4/30/02                   Bounds and Approximations                                                             Takehome final given


Last Day of Class!                                                                                              


5/7/02                     Final exam: In-class portion, 7:30-10:15                                            Takehome final due



OR 647 Course Coverage


Problem                                 What you need to know                                      What the computer does

Poisson probabilities           properties and simple calculations                    more extensive calculations


Erlang probabilities              properties and simple calculations                    more extensive calculations


Markov chains                     basic theory and solution to 3x3                       solution to more than 3x3


Linear difference eqns         formulation and solution to 3rd order                solution when order > 3


Prob. difference eqns          formulation and solution to 3rd order                solution when order > 3


Finite birth/death soln         approx. Nmax = 4                                  Nmax > 4


Infinite birth/death soln      soln when in closed form                                    complete soln


Little’s Law                           how to use                                                            automatic calculation


M/M/1                                   calculate all major measures                               --


M/M/c                                   c=2, all formulas                                                   c > 2


M/M/c/K                               c=1, all K, all formulas                                         c > 1


M/G/c/c                                 all G, c < 4                                                              c ³ 4


M/G/¥                                    all G                                                                        --


Finite Sources                       c = 1 only                                                               c > 1


Balking/Reneging                M/M/1 only, simple fns.                                     --


M[X]/M/1                               simple distrib for X                                              complete


M[K]/M/1                               K £ 3                                                                       K > 3


M/M[K]/1 (1):                         K = 2                                                                       K > 2


M/Ek/1                                    all k: P-K formula, etc.                                          everything else


Ek/M/1                                    as G/M/1, etc.


M/M/c Queues in Series     basic formulation; MOEs for c=1                      c > 1


Single-Server N/Ws             two nodes and all MOEs                                    more than 2 nodes


Multi-Server N/W                traffic eqns only                                                   remainder


Mean Value Analysis          traffic eqns only                                                   everything


M/G/1                                     P-K formula and related MOEs                          --


M/D/1                                    P-K formula and related MOEs                          complete solution


G/M/1                                     formulation & simple solution                           more complex solution


D/M/1                                    solution                                                                 --


G/G/1                                      simple transform analysis                                   --


M/D/c                                    simple transform analysis                                   remainder


G/G/c Bounds                       complete                                                                --


Heavy-Traffic Approxs.      complete


Design of Queues                                simple optimizations